Kernels and Multiple Windows for Estimation of the Wigner-Ville Spectrum of Gaussian Locally Stationary Processes

Abstract

Abstract-This paper treats estimation of the Wigner-Ville spectrum (WVS) of Gaussian continuous-time stochastic processes using Cohen's class of time-frequency representations of random signals. We study the minimum mean square error estimation kernel for locally stationary processes in Silverman's sense, and two modifications where we first allow chirp multiplication and then allow nonnegative linear combinations of covariances of the first kind. We also treat the equivalent multitaper estimation formulation and the associated problem of eigenvalue-eigenfunction decomposition of a certain Hermitian function. For a certain family of locally stationary processes which parametrizes the transition from stationarity to nonstationarity, the optimal windows are approximately dilated Hermite functions. We determine the optimal coefficients and the dilation factor for these functions as a function of the process family parameter.

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“…A common approach is to select the window based on minimizing the effective time-frequency area occupied by a given component [61], [62]. Other authors [46], [47], [53] and [63] even propose the use of separated kernels, with different parameters at different times. An open issue remains about how to measure the performance of the selected window.…”

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Optimal Scale Invariant Wigner Spectrum Estimation of Gaussian Locally Self-Similar Processes Using Hermite Functions

“…Several methods have been proposed for WVS estimation, such as: the multitaper method [2] and the mean square error (MSE) optimal kernel method [3], which is proposed for the class of Gaussian harmonizable processes. Moreover, in [4], it is shown that the WVS can be estimated as a weighted sum of spectrograms; also, the WVS can be approximated by a set of Hermite functions for the class of locally stationary processes. Such an approximation is advantageous when it comes to calculation, since only a limited number of Hermite functions need to be calculated [5].…”